FIG. 1 is a schematic diagram of a marine seismic survey in which seismic energy is emitted from a towed source 1 (e.g., an array of air gun strings) and detected by towed sensors (e.g., twin arrays 2, 2′ of streamers separated by a distance d, and each having multiple hydrophones S1, S2, . . . SN, and being suspended beneath floats/buoys 8) at a depth h below the surface 6 of a body of water. The source 1 imparts an acoustic wave to the water, creating a wavefield which travels coherently into the earth underlying the water. As the wavefield strikes interfaces 4 between earth formations, or strata, it is reflected back through the earth and water along a path 5 to the sensors, where it is converted to electrical signals and recorded.
In other marine survey methods, the sensors and/or sources are placed at or close to the seabed 3 or in wells (also called wellbores or boreholes) penetrating the earth formations. Through analysis of these detected signals, it is possible to determine the shape, position and lithology of the sub-sea formations.
A problem encountered in marine surveying, as well as in inverse vertical seismic profiling or “VSP,” is that of water column reverberation. The problem, which arises as a result of the inherent reflectivity of the water surface and bed (as well as sub-sea boundaries), may be explained as follows. A seismic wave reflected from seabed or sub-sea earth strata passes into the water in a generally upward direction. This wave, termed the “primary,” travels through the water and past the seismic sensors—whether on the seabed or in a towed array—which record its presence (i.e., characteristics of the primary). The wavefield continues upwardly, e.g., along path 7, to the water's surface, where it is reflected back downwardly. This reflected, or “ghost,” wavefield also travels through the water and past the sensor(s) where it is again recorded. Depending upon the nature of the earth material at the water's bottom, the ghost wavefield may itself be reflected upwardly through the water, giving rise to a series of one or more subsequent ghost reflections or “multiples.”
In instances where the earth material at the seabed is particularly hard, excess acoustic energy or noise generated by the seismic source can also become trapped in the water column, reverberating in the same manner as the reflected seismic waves themselves. This noise is often high in amplitude and, as a result, tends to cover the weaker seismic reflection signals sought for study. This reverberation of the seismic wavefield in the water obscures seismic data, amplifying certain frequencies and attenuating others, thereby making it difficult to analyze the underlying earth formations. Deghosting, or removal of the ghost wavefield(s), is therefore important for accurate characterization of earth formations. Those skilled in the relevant art will appreciate that deghosting alone does not entirely solve the multiple problem (although other known methods address multiples), since every multiple will have an up-going part as well as a down-going part (its ghost).
In most of the deghosting solutions proposed to date (e.g.: Robertsson, J. O. A., Kragh, J. E., and Martin, J., 1999, Method and system for reducing the effects of the sea surface ghost contamination in seismic data, GB Patent No. 2,363,459; Robertsson, J. O. A., and Kragh, J. E., 2002, Rough sea deghosting using a single streamer and a pressure gradient approximation, Geophysics, 67, 2005-2011; and Robertsson, J. O. A., Amundsen, L., Roesten, T., and Kragh, J. E., 2003, Rough-sea deghosting of seismic data using vertical particle velocity approximations, International Patent Application No. PCT/GB2003/002305, filed on 27 May 2003), three-dimensional (“3D”) effects are ignored. Data are assumed to be acquired with a source event, or “shot,” occurring in-line with a streamer, or otherwise to be pre-processed to satisfy this criterion.
However, in reality, 3D effects may be significant for several different reasons:                1. the acquisition geometry is 3D with significant cross-line offsets between some of the streamers and the source(s);        2. a 2D approach assumes cylindrical spreading of a wavefront in space, whereas in 3D it is assumed to be spherical;        3. the sea surface has a 3D structure causing scattering out-of-plane; and        4. there may be significant cross-line variation in the sub-surface causing out-of-plane reflections and scattering.        
In a “2D” approach such as the ones proposed in the past, we can successfully deal with reasons 1 and 2 above. In fact, with respect to reason 1, a compact deghosting filter (see, e.g., Robertsson and Kragh, 2002; Robertsson et al., 2003; and Roesten, T., Amundsen, L., Robertsson, J. O. A., and Kragh, E., Rough-sea deghosting using vertical particle velocity field approximations, 64th EAGE Conference Florens, 2002; or Amundsen, L., Roesten, T., Robertsson, J. O. A., and Kragh, E., On rough-sea deghosting of single streamer seismic data using pressure gradient approximations, submitted to Geophysics, 2003.) is ideally suited to project the actual plane of propagation onto the vertical plane containing the locations of the streamer data recordings without requiring irregularly spaced processing. Robertsson and Kragh (2002) showed how to compensate for reason 2 and concluded that the error made by assuming cylindrical spreading mostly is negligible. The two last items on the list (3 and 4) cannot be addressed using the “2D” approaches. Of these, the third item on the list, the 3D structure of the rough sea surface, is likely less important than the fourth item. Being able to properly account for wave propagation out-of plane is critical in areas with complex imaging tasks (salt, fault blocks, etc.) or multiple problems (e.g., diffracted multiples).
A need therefore exists for a solution to these shortcomings associated with 2D approaches.
The largest error term related to 3D effects in known 2D deghosting techniques (e.g., Robertsson and Krag, 2002; Robertsson et al., 2003; Amundsen et al., 2003) corresponds to a second-order cross-line spatial derivative of pressure. This could be implemented using a 3-point filter if data were acquired with three conventional streamers side-by-side spaced within a fraction of the Nyquist wavenumber (a few meters apart). For operational reasons, this is difficult achieve.
A need therefore exists for practical methods and/or apparatus that account for 3D effects when deghosting marine seismic data.